Transmission electron microscopy (TEM) and atomic force microscopy (AFM) have been used to analyse the core structure of dislocations in GaN grown by molecular beam epitaxy (MBE) as a function of the gallium to nitrogen ratio. Ga-rich samples had a much smoother morphology; TEM observations showed that amorphous deposits decorated some dislocations and occasional surface pits, but weak beam and end-on imaging suggested that, away from the growth surface, dislocations of all types had closed core structures, in contrast to previous observations (Hsu et al, Appl. Phys. Lett. 78, 3980 (2001), Baines et al, Mat. Res. Soc. Symp. Proc. 743, L2.5 (2003)). Ga-poor samples were found to have much rougher surfaces; dislocations were often at the centers of deep surface pits but were observed to be undecorated and to have closed core structures. It is concluded that in growth under Ga-rich conditions, decoration of dislocation cores depends on the accumulation of Ga at surface pits, rather than being a fundamental property of dislocation formation.